The Riemann -function is real on the 1/2-line and has all of its zeros there (assuming the Riemann Hypothesis). It is an entire function of order 1; because of its functional equation, is an entire function of order 1/2. It follows that the Riemann Hypothesis implies that all zeros of are on the 1/2-line. (See [ MR 84g:10070] for proofs of these statements.) Assuming the Riemann Hypothesis to be true, one can ask about the vertical distribution of zeros of , and more generally of . It seems that the zeros of higher derivatives will become more and more regularly spaced; can these distributions be expressed in a simple way using random matrix theory? See the unpublished paper [Differentiation evens out zero spacings] of David Farmer.
Back to the main index for L-functions and Random Matrix Theory.